(x-4)(2x+3)=x^2-4

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Solution for (x-4)(2x+3)=x^2-4 equation:


Simplifying
(x + -4)(2x + 3) = x2 + -4

Reorder the terms:
(-4 + x)(2x + 3) = x2 + -4

Reorder the terms:
(-4 + x)(3 + 2x) = x2 + -4

Multiply (-4 + x) * (3 + 2x)
(-4(3 + 2x) + x(3 + 2x)) = x2 + -4
((3 * -4 + 2x * -4) + x(3 + 2x)) = x2 + -4
((-12 + -8x) + x(3 + 2x)) = x2 + -4
(-12 + -8x + (3 * x + 2x * x)) = x2 + -4
(-12 + -8x + (3x + 2x2)) = x2 + -4

Combine like terms: -8x + 3x = -5x
(-12 + -5x + 2x2) = x2 + -4

Reorder the terms:
-12 + -5x + 2x2 = -4 + x2

Solving
-12 + -5x + 2x2 = -4 + x2

Solving for variable 'x'.

Reorder the terms:
-12 + 4 + -5x + 2x2 + -1x2 = -4 + x2 + 4 + -1x2

Combine like terms: -12 + 4 = -8
-8 + -5x + 2x2 + -1x2 = -4 + x2 + 4 + -1x2

Combine like terms: 2x2 + -1x2 = 1x2
-8 + -5x + 1x2 = -4 + x2 + 4 + -1x2

Reorder the terms:
-8 + -5x + 1x2 = -4 + 4 + x2 + -1x2

Combine like terms: -4 + 4 = 0
-8 + -5x + 1x2 = 0 + x2 + -1x2
-8 + -5x + 1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-8 + -5x + 1x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + -5x + 8 + x2 = 0 + 8

Reorder the terms:
-8 + 8 + -5x + x2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + -5x + x2 = 0 + 8
-5x + x2 = 0 + 8

Combine like terms: 0 + 8 = 8
-5x + x2 = 8

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = 8 + 6.25

Reorder the terms:
6.25 + -5x + x2 = 8 + 6.25

Combine like terms: 8 + 6.25 = 14.25
6.25 + -5x + x2 = 14.25

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 14.25

Calculate the square root of the right side: 3.774917218

Break this problem into two subproblems by setting 
(x + -2.5) equal to 3.774917218 and -3.774917218.

Subproblem 1

x + -2.5 = 3.774917218 Simplifying x + -2.5 = 3.774917218 Reorder the terms: -2.5 + x = 3.774917218 Solving -2.5 + x = 3.774917218 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = 3.774917218 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 3.774917218 + 2.5 x = 3.774917218 + 2.5 Combine like terms: 3.774917218 + 2.5 = 6.274917218 x = 6.274917218 Simplifying x = 6.274917218

Subproblem 2

x + -2.5 = -3.774917218 Simplifying x + -2.5 = -3.774917218 Reorder the terms: -2.5 + x = -3.774917218 Solving -2.5 + x = -3.774917218 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = -3.774917218 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -3.774917218 + 2.5 x = -3.774917218 + 2.5 Combine like terms: -3.774917218 + 2.5 = -1.274917218 x = -1.274917218 Simplifying x = -1.274917218

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.274917218, -1.274917218}

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